Wednesday, July 12, 2006

John Baez recently sparked quite a bit of discussion of music and mathematics over on sci.physics.research. But nobody seems to have mentioned the composer that many see as the most rigorously mathematical of all: Iannis Xenakis. Born in Romania with Greek ancestry and eventually adopted by the French, Xenakis started his career as a civil engineer and architect and only later turned to music.

I have listened to three CDs by Xenakis: Music for Strings, Persepolis and Legende D'Eer. Out of these, Music for Strings is a collection of pieces that come closest to the usual notion of music in that notes are played at various pitches on conventional instruments. Probably the most distinctive features are the wild glissandi flying in all directions (to use a spatial metaphor). The other two discs consist of hour long pieces that sound superficially like unpleasant extended accidents in a junkyard.

I decided to try to find out in what way these works were mathematical. After much searching on the web I found a paper by Edward Childs describing part of Xenakis's stochastic composition process. Apparently Xenakis made explicit use of four probability distributions in the composition of a piece call Achorripsis. I'll concentrate on three of these as I believe there is a way to drastically simplify Childs's description of these while combining them into one single scheme.

Xenakis composed his piece by creating a grid of 28 columns and 7 rows. Each row represents a group of instruments and each column represents a time period. Xenakis created a number of musical events and stochastically assigned these to cells in the grid. Within each grid cell he also chose the pitch of each event and the timing between events using stochastic methods. In particular, he generated the number of events in a cell using a Poisson distribution, the timing between events using an exponential distribution and the pitch of events using a uniform distribution. This composition dates from 1957 and this prompted Childs to say

The preparation of the score was a remarkable feat consideringthat he worked without the help of a computer, butcalculated all distributions, and their musical implementation,by hand.

Now, suppose that Xenakis had taken his grid and pinned it up on the wall. If he then stood some distance from the grid, blindfolded (actually, Xenakis would only have needed an eye patch), and had thrown darts at the grid, what distributions would we see? If he was far enough away and blindfolded there'd be unlikely to be any kind of bias towards one part of the grid or another meaning that the darts that hit the grid would be uniformly distributed. The number falling in each cell would have a Poission distribution. The spacing between successive pairs of darts along a horizontal axis would have an exponential distribution, and the heights of each dart would be uniformly distributed. In other words, Childs's description is entirely consistent with Xenakis having generated a large part of his composition with darts, with a single dart simultaneously generating the three random variables desired. So much for formalized music.

As far as Xenakis is concerned, let me emphasizeat once that I’d be much more interested in his researchif he hadn’t set out so obviously to reduce itsaccessibility and its credibility in a manner which isimmediately apparent as soon as you open his bookon formal musics.

Was Xenakis using mathematics to hide his composition methods from other, less mathematically savvy, composers?

Nonetheless, whatever his composition methods, after listening to Xenakis I seem to be finding other types of music to be far too clichéd and predictable. I think I have developed a liking for this composer. If you hear what seems to be the sound of pneumatic drills, thermal lances and scraping metal as you drive over the San Francisco Bay Bridge in the morning, look around, it might not be the bridge retrofit, but instead me commuting to work listening to one of Xenakis's electroacoustic works.

(PS If you're wondering, the fourth distribution, that I omitted, was the Maxwell-Boltzmann distribution, which Xenakis used to generate the speeds of the glissandi.)